Question

6x + 22 = A(x + 2) + B(x - 4)

^I am doing partial fraction decomposition and this is what I am at.

Attempt:
"Trick"
Let x = -2. Then,
6(2) + 22 = A(-2 + 2) + B(-2 - 4)
12 + 22 = 0 + B(-6)
34 = -6B
-34/6 = B

------
"No Trick"
6x + 22 = Ax + 2A + Bx - 4B
(LHS = RHS, then coeff's are equal),
6 = A + B, [1]
22 = 2A - 4B
=> 11 = A - 2B, [2a]

[2a] - [1],

5 = 0 -B
=> -5 = B.

Why aren't the two answers the same?

Answer 1

i am lost. where does this line come from
\[6x + 22 = Ax + 2A + Bx - 4B\]

Answer 2

oh nvm i see the mistake

Answer 3

6x + 22/(x+2)(x-4) dx

Answer 4

What is the mistake?

Answer 5

Let x = -2. Then, 6(2) + 22 = A(-2 + 2) + B(-2 - 4)

Answer 6

should be Let x = -2.
Then, 6(-2) + 22 = A(-2 + 2) + B(-2 - 4)

Answer 7

Ok I actually did it with -2, and then this is what I got:
6x + 22 = A(x + 2) + B(x-4).
Let x = -2. Then,

6(-2) + 22 = A(-2 + 2) + B(-2-4)
-12 + 22 = B(-6)
10/6 = -B
-10/6 = B....

Answer 8

6x+22=(a+b)x+(2a-4b)

now comparing both sides
6=a+b
22=(2a-4b)

now solve two equations you will get a and b

Answer 9

@SUROJ
I did that for "No Trick"

Answer 10

\[a+b=6\]
\[a+2b=11\]
\[b=5\]

Answer 11

What am I doing wrong wrt the trick? Why am I not getting the same answer as the second method?

Answer 12

(6x+22)/((x+2)(x-4))
Factor out the 2,
2(3x+11)/((x+2)(x-4))
----------------------------------------
A/(x+2)+B/(x-4)
----------------------------------------
6=A+B

22=-4A+2B
----------------------------------------
A=-5/3 B=23/3

-(5)/(3(x+2))+(23)/(3(x-4))

Answer 13

i see mistake when you put 6(2) in place of 6(-2) ....in first trick

Answer 14

did you see that?

Answer 15

ok lets start at the top
is it
\[\frac{6x + 22 }{(x+2)(x-4)}=\frac{a}{x+2}+\frac{b}{x-4}\]

Answer 16

Ok folks, the point here is that the "Trick" does not match with "No Trick" for the below solution:

6x + 22 = A(x + 2) + B(x-4).
Let x = -2. Then,

6(-2) + 22 = A(-2 + 2) + B(-2-4)
-12 + 22 = B(-6)
10/6 = -B
-10/6 = B....

Answer 17

@satellite73 Yes

Answer 18

Let x = -2. Then,
6(2) + 22 = A(-2 + 2) + B(-2 - 4) || in this step
12 + 22 = 0 + B(-6)
34 = -6B
-34/6 = B

Answer 19

@SUROJ Look at my recent one

Answer 20

that's right

Answer 21

Then why is it NOT the same as "No trick"?

Answer 22

ok so we can do this by writing
\[6x+22=A(x+2)+B(x-4)\]
\[-12+22=-6B\]
\[10=-6B\]
\[-\frac{5}{3}\] and likewise
\[A=\frac{23}{3}\] so give
\[\frac{6x+22}{(x+2)(x-4)}=\frac{23}{3(x+2)}-\frac{5}{3(x+4)}\]

Answer 23

a+b=6 is same as 2(a+b)=2(6) is same as 2a+2b=12 is same as -2a -2b =-12
add -2a-2b=-12 and 2a-4b=22

Answer 24

Alright, then how about the "No trick"?

Answer 25

typo on last line
should be
\[\frac{6x+22}{(x+2)(x-4)}=\frac{23}{3(x+2)}-\frac{5}{3(x-4)}\]

Answer 26

add left left and right right
-2a-2b=-12 and 2a-4b=22 gives -6b=10 so b is -10/6 same as first one

Answer 27

now we try with "no trick"
\[6x+22=A(x-4)+B(x+2)\]
\[6x+22=Ax-4A+Bx+2B\]
\[6x+22=(A+B)x+2B-4A\]
\[A+B=6\]
\[2B-4A=22\]

Answer 28

i get the same answer solving this one

Answer 29

yes.....